Values of $x$ and $y$ satisfying the equation ${\sin }^{7}y=|x^3-x^2-9x+9|+|x^3-x^2-4x+4|+{\sec }^{2}2y+{\cos }^{4}y$  are 
 

$$\begin{gathered} {\sin }^{7}y=|x^3-x^2-9x+9|+|x^3-x^2-4x+4|+{\sec }^{2}2y+{\cos }^{4}y \\ For x=1,{\sin }^{7}y={\sec }^{2}2y+{\cos }^{4}y \\ \Rightarrow {\sin }^{7}y{\cos }^{2}2y=1+{\cos }^{4}y{\cos }^{2}2y \\ Since, LHS \le 1. and RHS \ge 1. \\ which is possible only when \\ \therefore {\sin }^{7}y{\cos }^{2}2y=1 \\ \Rightarrow {\sin }^{7}y=1and{\cos }^{2}2y=1;y=\frac{\pi }{2} \end{gathered}$$

General value of $y$  is $2n\pi +\frac{\pi }{2}$ 
Hence, $x=1andy=2n\pi +\frac{\pi }{2},n\in I$